Mesoscopic polymer models

Roby and Joanny [27] improved the model of Benmouna, et al. [23] by incorporating interchain hydrodynamic interactions. At elevated concentrations, reptation dynamics were assumed, approximating the solution as a polymer melt in which mesoscopic polymer-solvent blobs are effective monomers. Hammouda [28] repeated the calculation, removing the restriction that the system contained equal amounts of two species having the same molecular weight, and analyzing tagged-tracer experiments. 

An extensive hterature is devoted to the physics of water channels formation and to the mechanism of proton and water transport in membranes. In this section, we give a brief overview relevant to the analytical modelling of fuel cells. A detailed review of phenomenological membrane transport models is given in (Weber and Newman, 2007). Atomistic modelling and experiments on proton transport in membranes are reviewed in (Kreuer et al., 2004). Recent advances in mesoscopic membrane modelling are discussed in (Promislow and Wetton, 2009). The reader is referred to these works for a detailed discussion of the transport processes in polymer membranes. 

In contrast to D, the prediction of other viscoelastic properties, such as the friction coefficient f or the zero-shear rate viscosity i/o, requires that the atomistic MD data be mapped upon a mesoscopic theoretical model. For unentangled polymer melts, such a model is the Rouse model, wherein a chain is envisioned as a set of Brownian particles connected by harmonic springs [25,28]. For entangled polymer melts, a better model that describes more accurately their dynamics is the tube or reptation model [26]. According to this model, the motion of an individual chain is restricted by the surrounding chains within a tube defined by the overall chain contour or primitive path. During the lifetime of this tube, any lateral motion of the chain is quenched. 

Usually, the mesoscopic, kinetic models are considered to be well suited for predicting dynamic properties of polymer solutions on macroscopic scales. Details of the fast solvent dynamics are in most cases irrelevant for macroscopic properties. Exceptions are polyelectrolytes, where the motion of counterions in the solvent can have a major influence on polymer conformation. Therefore, more microscopic models of polyelectrolytes with explicit counterions are sometimes employed [34] (see also the contribution by M. Muthukumar in this volume). Another exception is the dynamics of individual biopolymers, for example, protein folding, which is modeled with an all atomistic model including an explicit treatment of the (water) solvent molecules [35]. 

For the numerical simulation of flowing polymers, several mesoscopic models have been proposed in the last few years that describe polymer (hydro-)dynamics on a mesoscopic scale of several micrometers, typically. Among these methods, we like to mention dissipative particle dynamics (DPD) [168], stochastic rotation dynamics (sometimes also called multipartide collision dynamics) [33], and lattice Boltzmann algorithms [30]. Hybrid simulation schemes for polymer solutions have been developed recenfly, combining these methods for solvent dynamics with standard particle simulations of polymer beads (see Refs [32, 169, 170]). Extending the mesoscopic fluid models to nonideal fluids including polymer melts is currently in progress [30, 159,160,171]. 

The hierarchy of models is complemented by a variety of methods and tecluiiques. Mesoscopic models tliat incorporate some fluid-like packing (e.g., spring-bead models for polymer solutions) are investigated by Monte Carlo... 

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic stmcture of the chemical repeat units [4, 5 and 6]. The self-similar stmcture [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

The vector field entirely and uniquely determines the stream lines and their properties. As we focus our attention on the mesoscopic properties of stream lines, assuming that they can resemble a polymer-like amorphous packing of chain backbones, we have to consider in greater detail their intrinsic properties. As shown in the next section, Santos and Suter [98] elaborated a model for generating packing structures of Porod-Kratky chains. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

The problem of linking atomic scale descriptions to continuum descriptions is also a nontrivial one. We will emphasize here that the problem cannot be solved by heroic extensions of the size of molecular dynamics simulations to millions of particles and that this is actually unnecessary. Here we will describe the use of atomic scale calculations for fixing boundary conditions for continuum descriptions in the context of the modeling of static structure (capacitance) and outer shell electron transfer. Though we believe that more can be done with these approaches, several kinds of electrochemical problems—for example, those associated with corrosion phenomena and both inorganic and biological polymers—will require approaches that take into account further intermediate mesoscopic scales. There is less progress to report here, and our discussion will be brief. 

Mesoscopic Modeling of Two-Phase Transport in Polymer Electrolyte Fuel Cells... 

The second contribution spans an even larger range of length and times scales. Two benchmark examples illustrate the design approach polymer electrolyte fuel cells and hard disk drive (HDD) systems. In the current HDDs, the read/write head flies about 6.5 nm above the surface via the air bearing design. Multi-scale modeling tools include quantum mechanical (i.e., density functional theory (DFT)), atomistic (i.e., Monte Carlo (MC) and molecular dynamics (MD)), mesoscopic (i.e., dissipative particle dynamics (DPD) and lattice Boltzmann method (LBM)), and macroscopic (i.e., LBM, computational fluid mechanics, and system optimization) levels. 

Miiller-Plathe F (2002) Coarse-graining in polymer simulation From the atomistic to the mesoscopic scale and back. J Chem Phys Phys Chem 3 754—769 Muller R, Picot C, Zang YH, Froelich D (1990) Polymer chain conformation in the melt during steady elongational flow as measured by SANS. Temporary network model. Macromolecules 23(9) 2577—2582... 

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